Maximum Spectral Measures of Risk with Given Risk Factor Marginal Distributions

26 Pages Posted: 2 Dec 2020

See all articles by Mario Ghossoub

Mario Ghossoub

University of Waterloo

Jesse Hall

University of Waterloo

David Saunders

University of Waterloo

Date Written: October 23, 2020

Abstract

We consider the problem of determining an upper bound for the value of a spectral risk measure of a loss that is a general nonlinear function of two factors whose marginal distributions are known, but whose joint distribution is unknown. The factors may take values in complete separable metric spaces. We introduce the notion of Maximum Spectral Measure (MSP), as a worst-case spectral risk measure of the loss with respect to the dependence between the factors. The MSP admits a formulation as a solution to an optimization problem that has the same constraint set as the optimal transport problem, but with a more general objective function. We present results analogous to the Kantorovich duality, and we investigate the continuity properties of the optimal value function and optimal solution set with respect to perturbation of the marginal distributions. Additionally, we provide an asymptotic result characterizing the limiting distribution of the optimal value function when the factor distributions are simulated from finite sample spaces. The special case of Expected Shortfall and the resulting Maximum Expected Shortfall is also examined.

Keywords: Spectral risk measure, Expected shortfall, Dependence uncertainty, Optimal transport, Monge-Kantorovich duality.

JEL Classification: C02, C61, G21, G22.

Suggested Citation

Ghossoub, Mario and Hall, Jesse and Saunders, David, Maximum Spectral Measures of Risk with Given Risk Factor Marginal Distributions (October 23, 2020). Available at SSRN: https://ssrn.com/abstract=3720332 or http://dx.doi.org/10.2139/ssrn.3720332

Mario Ghossoub

University of Waterloo ( email )

Dept. of Statistics & Actuarial Science
200 University Ave. W.
Waterloo, Ontario N2L 3G1
Canada

HOME PAGE: http://uwaterloo.ca/scholar/mghossou

Jesse Hall

University of Waterloo ( email )

David Saunders (Contact Author)

University of Waterloo ( email )

200 University Avenue West
Waterloo, Ontario N2L 3G1
Canada
(519) 888-4567 (Phone)

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